I have started a great many games over the last 25 years, but have finished about a quarter of them. Typically, one of two events will take place. I will get stuck, which leads me to become frustrated and move on to a different game. Or, I will get a new game, which grabs my attention and causes any previously played and unfinished game to remain in that state.
The games I have finished are almost all RPGs:
The SSI Gold Box D&D games
Baldur's Gate 1
Baldur's Gate 2
Planescape: Torment
Icewind Dale I
Icewind Dale 2
Oblivion (but not Morrowind, although I have spent over 200 hours playing it and all of the expansions)
Vampire the Masquerade: Bloodlines
Knights of the Old Republic
Knights of the Old Republic 2
Jade Empire
Neverwinter Nights (I have two fights left in NWN2 that I never get around to finishing)
Mass Effect
Crackdown
Lionheart
Sacred
Diablo (which I do not count as an RPG, but as an action game with a few RPG elements)
I have easily ten times as many unfinished games as ones I've finished, which is why I have stopped buying newly released titles. I'd like to finish some of the games I already have before buying any more. The only exception I am willing to make at this time is for Fallout 3, which I plan to pick up this week, even though I've not finished the first one and have never played, but own, the second one.

This list is longer than I want it to be;
GTAIV
System Shock 2
Fallout
Mass Effect
Deus Ex
Max Payne 2
Yakuza 2

The Witcher EE - Bugs, people hard to find, monsters in the swamp are too hard to kill (the worm things that come out of the ground and spit/throw something at you.), I hate not being able to get my sword out in the day time (made the killing the dogs quest annoying. "Oh look a dog! Oh wait it's daytime... zzzz").
Roller coaster tycoon 3 - Making crazy rollercoasters is fun. Managing a theme park is not.
Freespace 2 - It kept crashing with my 8800GT. Got a ATI 4870X2 a few weeks back and now when I load it up my profile isn't there :(
X3: Reunion - I literally only played 10-20 minutes before realising I didn't have a clue what to do. I shouldn't have to read tutorials to play a game...
Tribes vengeance - I got to the driving level, then got bored. I did try playing it again last year but it was boring.
UT3 - I just play the quick matches. And always instagib.
Ground control - It came in a buy 2 get 1 free deal. I never really liked strategies though.

Lets see...
Tomb Raider - The Last Revelation
Blood Omen - Legacy of Kain
Hitman 2
Hitman: Contracts
X-Com: Interceptor
Mechwarrior 4: Vengeance
American McGee's Scrapland
Transformers - The Game
Jedi Knight - Jedi Outcast
Jedi Knight - Jedi Academy
Grand Theft Auto 3
Neverwinter Nights
Thief Deadly Shadows
The sad part is, I still play all of these occasionally (at least the ones that still work in XP), but I still never seem to be able to finish any of them.
EDIT - Looks like I can take one off of that list. I just played a little Scrapland tonight and was shocked to discover that I was only two or three missions away from completing it when I last played it months ago. I finally finished the main story, but I do still need to complete some of the side missions.

There was an article on PC Zone, I think, a couple years ago, describing different types of gamers. The No.1 was a group called 'grazers' that always have their toes in multiple games at the same time.
I am a grazer. So the fact there are games that currently I haven't finished doesn't make them bad games, it just says a lot about how I play games!

Well, let's see...
The Darkness (finished it once, decided that the other achievements just weren't worth it.)
Final Fantasy V (nope, no, 0/10, get out of my DS. I dunno what it really was about V, I just hated it.)
Metroid Prime 3 (Beat 1 and 2, but the backtracking at the end caused me to tear this game from my Wii and burn it as though it were plague ridden.)
Willow (anyone remember this one? I played it a while back on my NES emulator since the cartridge got sold in a yard sale, I don't have the drive to finish this one, not bad, but not enough motivation.)
Halo 2 (It's just like all other FPS games, the game is honestly a piece of garbage if you look past the setting and ignore the thirteen year olds parroting how great Master Chief is.)
Two Worlds (Played for three hours [one hour a day for three days] then torn from my Xbox and sold like the garbage it is.)
Oblivion (two playthroughs with next to no deaths on the hardest setting, yeah, I'm done.)
Age of Empires III (the online is filled with idiots who get upset over walls built around your settlement. That, and the gameplay is a tad slow, especially compared to other RTS games.)
Assassin's Creed (Beat it once, I am NOT watching the damn credits every time I want to play the game.)
Vagrant Story (Actually, I found this at a local Wal-mart for five bucks about two weeks back, it's still in the wrapping. It kinda got overshadowed by other games.)
The list goes on, it happens all the time, I guess.

RPG's usually account for my piles of shame (the Squaresoft variety in particular). I really should stop wasting my time on these. But I'll beat you someday, Final Fantasy Tactics! Right when I'm done with Fallout, FFXII, Chrono Cross...Ah screw the whole thing.

RTS for me. I don't like them. Mainly because I just like controlling one character, so fps/acton, rpg and mmos are for me. I don't think you'll get any mmos on here though.

Wrong.
Horizons. Pure crap. It had great potential, but pure crap. Mostly due to Artifact Entertainment being idiots. Also focusing ALL their attention on the dragon race and ignoring the 'bipeds' was stupid. Stupid resources distribution too.
Vanguard. Past level 20 it was virtually impossible to be without a group. Plus they made 'rare' resources too common.
Pirates of the Burning Sea. While it had absolutely great ship to ship combat, that was all the game had. Just not enough to keep me interested.
Age of Conan. I didn't even make it out of the starting city. Waiting til level 40 to craft? A friend of mine who power games quit a few days after I did, because of no endgame. I heard they are already merging servers.
I'm hoping Darkfall will be good, but since there really is no info other than generalities on their web site, and its supposed to be done beta in 2 months, I really am not going to pin a lot of hopes on this one.

i have a bad bad habit of going ooo shiny new game! so the list begins
Disgaea
Chaos Legion
Devil May Cry 1& 4
.hack
Ico
Okami
Phantom Brave
Shadow Hearts: From the new World
SMT: Nocturne
SMT: Digital Devil Saga 2
SMT: Devil Summoner: Raidou Kuzunoha Vs the soulless army
SMT: Persona 1 & 3
Star Ocean 2 & 3
Xenosaga 2
Baulders gate 2
Neverwinter Nights 1&2
Vampire the Masquerade: Bloodlines
Morrowind
command and conquer 3
C&C: Renegade
Shogo
Aliens vs Predator 2 (useless humans)
Starlancer
Freelancer
One must Fall Battlegrounds
Jade Empire
that's all off the top of my head i know there's more somewhere

A TON of strategy games that get really hard towards the end.
Einlanzer series.
X-Com series.
Command & Conquer series.
Games that are too repetitive.
Summon Night.
Diablo I.
Final Fantasy VIII.
Golden Sun.
Games where I just don't care anymore.
Morrowind.
Oblivion.
Mega Man Battle Network.
Resident Evil.
Front Mission series. (They can be really hit-and-miss.)
Games that I start, but look so hideous or have some other major flaw that keeps me from playing further.
Far Cry.
IL2 Strumovik.
Deus Ex.
Vagrant Story. Eternal Eyes. Both pretty mundane RPGs on the PSX.

I bought Monster Madness a couple of months ago and only beat the first level since a friend let me borrow Bioshock and that took over. Also I bought Persona 3 but never opened it since my sister decided she wanted to have the ps2.

Okay, first up the last rebel mission in tachyon where you face down the corp heavy cruiser made more ridiculous by how easy the last corp mission is.
Second, Project IGI. A game i love but having overlooked the no sve feature for the whole game they throw wave after wave of goons at you all while asking you to defend your useless team-mate in the longest level of the game.
And Laslty "No one lives forever": all downhill after the first level. i just didnt care anymore and Im a serious gamer.

My small list of games (on the closest shelf) that I have yet to complete:
S.T.A.L.K.E.R. Shadow of Chernobyl
Painkiller
Rayman Raving Rabbids
Sacred
SpellForce
Hitman: Codename 47 and Bloodmoney (completed Hitman 2 though)
Age of Wonders
O.R.B
Titan Quest: Immortal Throne
NeverWinter Nights
Call of Cthulhu: Dark Corners of the Earth
Scrapland
Will Rock
Enclave
The Witcher
Perimeter: Emperor's Testament
Fallout 2 (completed FO1 several times though)
Universe at War: Earth Assault
Monster Madness: Battle for Suburbia
Galactic Civilizations
Emperor: Battle for Dune
Z: Steel Soldiers
Metroid Prime 3: Corruption
Super Mario Galaxy
Sonic 3D
Baldur's Gate 2
The Temple of Elemental Evil (constant crashing did that game in for me, I'd have completed it otherwise)
Some of them (like Rayman and Baldur's Gate 2) won't be completed by me ever due to their gameplay, others (like Metroid Prime 3) just need the right time and place. It is less than half the games on that shelf though.

I wasn't wrong at all. What I meant by they won't get any mmos on here is, this site itself won't be selling them. Not that people would be listing them.
Yeah Age of Conan is horrible. I disliked it for different reasons, game development issues really. I wrote a big review about that one and Warhammer (not on this site.)
I love City of Villains though, and the newer game itself is a lot better than how CoH was 3 years ago.
Vanguard would have been okay and I could have even overlooked the kind of ugly player models and unimaginitive armor, if they just let people solo the hardest stuff, which is my biggest dislike for mmos. I don't want to argue about mmos "needing" to be group based since it's a MMO after all, but I miss the old days for some mmos, when some classes could solo the hardest stuff or at least duo it. Duo is good. More people seems non-challenging or boring for my playstyle. Like those big raids in WoW, it becomes less a personal challenge and more of a repetitious, mindless act you perform with a bunch of people for items. To me that takes all the fun and tosses it out of the mix.
Horizons was good in the very beginning but it went bad. I liked aspects of that game a lot. It just had so many bugs and as you said focusing on the dragon race. They didn't focus too much on that though when I played. The game looked a lot different in it's early months to maybe half a year, than it did a year later from release. My Elemental Archer was level 118 in the first 4 weeks of the game, due to a nice log out bug to recycle one of the skills that turned the bow into a machine gun. I don't recall the name.
I hope Darkfall is good. It seems they are taking a lot of tips from how UO was in the old days (before trammel/facets/item based play) which to me is still the best mmo (until Sunsword and EA games got involved.) Skill based mmos are better than level based in my opinion.
Anyway, I mostly play FPS games or RPGs.
Oh and Hey, thanks DeathKitten!